Carryborrow

Carryborrow is a term sometimes used in discussions of positional numeral systems to describe the paired operations of carrying in addition and borrowing in subtraction. It is not a formal mathematical object with a single precise definition, but rather a pedagogical shorthand for the mechanics that occur when digits in successive places interact under base-n arithmetic.

For example, in base 10, 47 plus 58 demonstrates carrying: units column 7 + 8 = 15, write 5 and carry 1 to the tens column; tens column 4 + 5 + 1 = 10, write 0 and carry 1 to the hundreds column; the final result is 105.

Subtraction also involves borrowing. For instance, 52 minus 19 shows borrowing: units column 2 - 9 cannot be done without borrowing; borrow 1 from the tens place to make 12 - 9 = 3; tens column now 4 - 1 = 3; so the result is 33.

In computer arithmetic, carry and borrow correspond to carry-out and borrow-out signals, used to link consecutive digit operations. They influence how multi-digit results are formed and how subsequent instructions or operations are scheduled. Some educational treatments present them together under the label carryborrow to emphasize their relationship in base-n arithmetic.

Because carry and borrow are complementary operations, the term carryborrow is most common in teaching materials and informal explanations, rather than as a formal concept in higher mathematics. It serves to highlight the shared mechanism that governs how digits interact across positions during addition and subtraction.